Boundedness and Stability in a Chemotaxis-Growth Model with Indirect Attractant Production and Signal-Dependent Sensitivity

Abstract

We study the chemotaxis-growth system with signal-dependent sensitivity function and logistic source

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\nabla \cdot \bigl(u\chi (v)\nabla v\bigr)+\mu u(1-u), &x \in \varOmega ,\ t>0, \\ v_{t}=d\Delta v+h(v,w), &x\in \varOmega ,\ t>0, \\ \tau w_{t}=-\delta w+u, &x\in \varOmega ,\ t>0, \\ \end{array}\displaystyle \right . \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\varOmega \subset \mathbb{R}^{n}\ (n\geq 1)$, where the parameters $\mu , \tau , \delta >0$ and $d\geq 0$, the functions $\chi (v)$, $h(v,w)$ satisfying some conditions represent the chemotactic sensitivity and the balance between the production and degradation of the chemical signal which relies explicitly on the living organisms, respectively. In the case that $\chi (v)\equiv 1$, $d=1$ and $h(v,w)=-v+w$, Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) asserted global existence of bounded solutions for arbitrary $\mu >0$ and established asymptotic behavior of solutions to the mentioned system under the condition $\mu >\frac{1}{8\delta ^{2}}$ in the three dimensional space. The purpose of the present paper is to investigate the global existence and boundedness of classical solutions and to improve the condition assumed in Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) by extending the previous method for obtaining asymptotic stability. Consequently, the range of $\mu$ is extended in the present paper.